A mapping from the vertex set of a graph G=(V,E) into an interval of integers
{0,...,k} is an L(2,1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2
apart, and every two vertices with a common neighbour are mapped onto distinct integers. It is known that for any fixed k>=4, deciding
the existence of such a labelling is an NP-complete problem while it is polynomial for k<=3. For even k>=8, it remains NP-complete when restricted to planar graphs. In this paper, we show that it remains NP-complete for any k>=4 by reduction from Planar Cubic Two-Colourable Perfect Matching. Schaefer stated without proof that Planar Cubic Two-Colourable Perfect Matching is NP-complete.
In this paper we give a proof of this.